a wire of length 4m and radius 1 mm has a resistance of 1 ohm.what length of the wire of the same material at the same temperature and of radius 2 mm will also have a resistance of 1 ohm
To find the length of the wire with a radius of 2 mm that will have a resistance of 1 ohm, we can use the formula for the resistance of a wire:
R = (ρ * L) / A,
where R is the resistance, ρ (rho) is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
Now, the resistance of the wire is given as 1 ohm, so we have:
1 = (ρ * L) / A.
Since we want to find the length of wire when the radius is doubled (2 mm), we can calculate the new cross-sectional area using the formula for the area of a circle:
A = π * r^2,
where r is the radius of the wire.
Let's substitute the values into the formula:
A = π * (2 mm)^2 = π * 4 mm^2.
Now we need to find the resistivity of the material. Assuming it remains the same, we don't need to calculate it. We can cancel it out in the equation.
Let's rearrange the formula to solve for the length of the wire:
L = (R * A) / ρ.
Substituting the values:
L = (1 ohm * π * 4 mm^2) / ρ.
Since ρ is the same for both wires, we can assume it cancels out, and we are left with:
L = 4π mm^2 / ρ.
Therefore, the length of the wire with a radius of 2 mm and a resistance of 1 ohm is 4π mm^2 divided by the resistivity of the material.